Other Angle Relationships in Circles

Section 10.4

Goal:

- To solve problems using angles formed by tangents, chords and lines that intersect a circle.

Theorem 10.12

C

A

B

2

1 11 and 2

2 2m mAB m mBCA

If m 1=80 , then ?

If 200, then m 2=?

mBA

mBCA

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

1

Example Find: m CBD and mBAD

AB

4 50x 3x

D

C

Lines Intersecting On, Inside, or Outside a Circle

On the circle Inside the circle Outside the circle

Case 1 Case 2 Case 3

An inscribed angle

Lines, or chords, intersecting inside a circle

Theorem 10.13

1

B

A

C

11

2m mCD mAB

D

2

If two chords intersect in the interior of a circle, then the measures of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

12

2m mBC mAD

Chords, intersecting inside a circle Example

x

B

A

C

Find x

D

40

120

Lines Intersecting Outside a CircleThree scenarios!

tangent-secant tangent-tangent secant-secant

Tangent - Secant

1

B

A

C

Tangent-Secant

11

2m mBC mAC

1 210 and 40Find m if mBC mAC

Tangent - Tangent

1

Q

P

R

Tangent-Tangent

11

2m mPQR mPR

1 300 and 60Find m if mPQR mPR

Secant - Secant

1

B

A

D

Secant-Secant

11

2m mBC mAD

1 190 and 30Find m if mBC mAD

C

Example

78

B

A

C

Find x:

204

x

Example

96

Find x:

x

Example

31

Find x:

57

x

Example

27

Find x:

88

x